Is it possible to set the image of the image texture to be a camera in that scene?
You'll need to do some compositing to make this work, because you can only have one active (rendering) camera per scene. So yes, you will have to render both, but Blender can auto-composite your results.

Make a new scene. You can do a Full Copy if you want to copy everything over, and delete all but the mouth and the camera. Keep the background camera as perspective, and make the camera in the new scene orthographic. In the new scene with the mouth, make sure you have Film -> Transparent checked. You might want to copy the lights over as well.

In the compositor, check "Use nodes," and duplicate the existing Render Layers node. Change its scene, so that both scenes are present. Connect the two nodes' image outputs to an Alpha Over node, which will layer them. Connect that to the composite output, or the viewer output if you'd like to preview what's going on. Look up some compositor tutorials if you're not sure what all this means.